Abstract
The character theory of finite groups has numerous basic questions that are often already quite involved: enumeration of irreducible characters, their character formulas, point-wise product decompositions, and restriction/induction between groups. A supercharacter theory is a framework for simplifying the character theory of a finite group, while ideally not losing all the important information. This paper studies one such theory that straddles the gap between retaining valuable group information while reducing the above fundamental questions to more combinatorial lattice constructions.
Highlights
Through the work of André [3] and Yan [9], supercharacter theory has allowed us to apply the tools of character theory to groups previously deemed intractable
It turns out that every supercharacter theory identifies a sublattice of normal subgroups, and this naturally partitions supercharacter theories according to the sublattices they “see.” Aliniaeifard identified the unique coarsest supercharacter theory corresponding to each sublattice, and identified numerous desirable characteristics exhibited by this theory
This point of view naturally leads to a notion of “simple” supercharacter theory, or one which only identifies the trivial subgroup and the whole group
Summary
Through the work of André [3] and Yan [9], supercharacter theory has allowed us to apply the tools of character theory to groups previously deemed intractable. It turns out that every supercharacter theory identifies a sublattice of normal subgroups, and this naturally partitions supercharacter theories according to the sublattices they “see.” Aliniaeifard identified the unique coarsest supercharacter theory corresponding to each sublattice, and identified numerous desirable characteristics exhibited by this theory This point of view naturally leads to a notion of “simple” supercharacter theory, or one which only identifies the trivial subgroup and the whole group (e.g. simple groups only have simple supercharacter theories). An advantage of the normal lattice supercharacter theories is that they are somewhat more canonical, analogous to how every group has a partition by conjugacy classes. This feature allows us to better compare supercharacter theories of groups related via inclusion. In this paper we instead illustrate the theory with several families of abelian groups including cyclic groups and vector spaces
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